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几何代数GA_Lecture4

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几何代数GA_Lecture4GeometricAlgebraDrChrisDoranARMResearch4.AlgebraicFoundationsand4DAxiomsL4SElementsofageometricalgebraarecalledmultivectorsSpaceislinearoverthescalars.AllsimpleandnaturalMultivectorscanbeclassifiedbygradeGrade-0termsarerealscalarsGradingisaprojectionoperationA...

几何代数GA_Lecture4
GeometricAlgebraDrChrisDoranARMResearch4.AlgebraicFoundationsand4DAxiomsL4SElementsofageometricalgebraarecalledmultivectorsSpaceislinearoverthescalars.AllsimpleandnaturalMultivectorscanbeclassifiedbygradeGrade-0termsarerealscalarsGradingisaprojectionoperationAxiomsL4SThegrade-1elementsofageometricalgebraarecalledvectorsSowedefineTheantisymmetricproduceofrvectorsresultsinagrade-rbladeCallthistheouterproductSumoverallpermutationswithepsilon+1forevenand-1foroddSimplifyingresultL4SGivenasetoflinearly-independentvectorsWecanfindasetofanti-commutingvectorssuchthatSymmetricmatrixDefineThesevectorsallanti-commuteThemagnitudeoftheproductisalsocorrectDecomposingproductsL4SMakerepeateduseofDefinetheinnerproductofavectorandabivectorGeneralresultL4SOver-checkmeansthistermismissingGrader-1Definetheinnerproductofavectorandagrade-rtermRemainingtermistheouterproductCanprovethisisthesameasearlierdefinitionoftheouterproductGeneralproductL4SExtenddotandwedgesymbolsforhomogenousmultivectorsThedefinitionoftheouterproductisconsistentwiththeearlierdefinition(requiressomeproof).Thisversionallowsaquickproofofassociativity:Reverse,scalarproductandcommutatorL4SThereverse,sometimeswrittenwithadaggerUsefulsequenceWritethescalarproductasScalarproductissymmetricOccasionallyusethecommutatorproductUsefulpropertyisthatthecommutatorwithabivectorBpreservesgradeRotationsL4SCombinationofrotationsSotheproductrotorisRotorsformagroupSupposewenowrotateabladeSothebladerotatesasFermions?TakearotatedvectorthroughafurtherrotationTherotortransformationlawisThisisthedefiningpropertyofafermion!UnificationL4SOneofthedefiningpropertiesofspin-halfparticlesdropsoutnaturallyfromthepropertiesofrotors.LinearalgebraL4SLinearfunctionfExtendftomultivectorsThisagrade-preservinglinearfunctionThepseudoscalarisuniqueuptoscalesowecandefineFormtheproductfunctionfgQuicklyprovethefundamentalresultProjectivegeometryL4SUseprojectivegeometrytoemphasiseexpressionsinGAhavemultipleinterpretationsClosertoGrassmann’soriginalviewOurfirstapplicationof4DGACoretomanygraphicsalgorithms,thoughrarelytaughtProjectivelineL4SPointxrepresentedbyhomogeneouscoordinatesApointasavectorinaGAOuterproductoftwopointsrepresentsalineDistancebetweenthepointsScalefactorsThisrepresentationofpointsishomogeneousCrossratioL4SDistancebetweenpointsInvariantquantityCanseethattheRHSisinvariantunderagenerallineartransformationofthe4pointsRatioisinvariantunderrotations,translationsandscalingProjectiveplaneL4SPointsonaplanerepresentedbyvectorsina3DGA.Typicallyalignthe3axisperpendiculartotheplane,butthisisarbitraryPointLinePlaneInterchangepointsandlinesbyduality.Denoted*Intersection(meet)definedbyFor2linesFor3linestomeetatapointReducestosimplestatementExampleL4SCanprovethealgebraicidentityThese3pointsarecollineariffthese3linesmeetatapointThisisDesarguestheorem.AcomplexgeometricidentityfrommanipulatingGAelements.Projectivegeometryof3DspaceL4SPointLinePlaneInterchangepointsandplanesbyduality.LinestransformtootherlinesVolumeIn4DwecandefinetheobjectThisishomogenous,butNOTablade.AlsosatisfiesBivectorsforma6dimensionalspaceBladesrepresentlinesTestofintersectionis2Bivectorswithnon-vanishingouterproductare2linesmissingeachotherPluckercoordinatesandintersectionL4SConditionthatabivectorBrepresentsalineisWritePlucker’sconditionAlinearrepresentationofaline,withanon-linearconstraintSupposewewanttointersectthelineLwiththeplanePResourcesL4Sgeometry.mrao.cam.ac.ukchris.doran@arm.comcjld1@cam.ac.uk@chrisjldoran#geometricalgebragithub.com/ga
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